Finite Group Databasebeta

← #6463628

Subgroups — #6463628

Order 512.

1930 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34275C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic, complemented
3#29267C214yesmaximal, complemented
4#29263C214yesmaximal, complemented
5#29261C214yesmaximal, complemented
6#29256C214yesmaximal, complemented
7#23733C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic, complemented
9#23454C211yesmaximal, characteristic, complemented
10#23451C211yesmaximal, characteristic, complemented
11#21263C211yesmaximal, characteristic
12#21263C211yesmaximal, characteristic
13#21174C211yesmaximal, characteristic, complemented
14#21173C211yesmaximal, characteristic, complemented
15#20882C211yesmaximal, characteristic, complemented
16#20881C211yesmaximal, characteristic, complemented
17#20876C211yesmaximal, characteristic, complemented
18#20876C211yesmaximal, characteristic, complemented
19#20754C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#424C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#417C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C2×#416C2211yescharacteristic
35C2×#416C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38#2368—24no—
39#2363—22no—
40#2361—22no—
41#2358—24no—
42#2355—22no—
43#2355—22no—
44#2331C2214yescomplemented
45#2328C2212yes—
46#2326C2212yes—
47#2317C2211yescharacteristic
48#2317C2211yescharacteristic
49#2317C2211yescharacteristic
50C4×#127C2212yes—
51C4×#125C2214yescomplemented
52C4×#124C2212yes—
53#2297C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#382C2211yescharacteristic
56#2274C2211yescharacteristic
57#2274C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2097C2212yescomplemented
60#2097C2212yescomplemented
61#2092C2214yes—
62#2092C2214yes—
63#2092C2212yescomplemented
64#2092C2212yescomplemented
65#2026C2212yescomplemented
66#2025C2214yes—
67#2025C2212yescomplemented
68#2024C2212yescomplemented
69#2023C2214yes—
70#2023C2212yescomplemented
71#1983C2214yes—
72#1983C2214yes—
73#1982C2214yes—
74#1982C2214yes—
75#1949C2214yescomplemented
76#1949C2214yescomplemented
77#1948C2214yescomplemented
78#1947C2214yescomplemented
79#1865C2214yescomplemented
80#1864C2214yes—
81#1864C2214yes—
82#1864C2214yescomplemented
83#1863C2214yescomplemented
84#1862C2214yes—
85#1862C2214yes—
86#1862C2214yescomplemented
87#1832C2214yescomplemented
88#1831C2214yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#535—24no—
104#535Q818yescomplemented
105#533—24no—
106#533—24no—
107C2×C4⋊Q8C2312yes—
108C2×#127C2312yes—
109C2×#125C2312yes—
110C2×#124C2312yes—
111C2×C22⋊Q8—22no—
112C2×#121—22no—
113C2×#120—22no—
114#520—42no—
115#520—42no—
116#518—42no—
117#518—42no—
118C2×C4×Q8—22no—
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4—24no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×#117—21no—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—22no—
138C22×C22⋊C4—22no—
139C22×C42C2311yescharacteristic
140#428—42no—
141#428—42no—
142C2×#107—42no—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#106—42no—
147C2×#106C2314yes—
148C2×#106C2312yes—
149C2×#106C2312yes—
150#424—24no—
151#424—24no—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154#418—24no—
155#418—24no—
156#417—42no—
157#417—42no—
158#416—24no—
159#416—24no—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#102—42no—
165C2×#102—42no—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174#391—24no—
175#391—24no—
176#391C2311yescharacteristic
177#391C2311yescharacteristic
178#390—24no—
179#390—22no—
180#390C2311yescharacteristic
181#390C2312yes—
182#390C2312yes—
183#390C2311yescharacteristic
184#389—22no—
185#389C2314yes—
186#389C2314yes—
187#388—24no—
188#388—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#387—24no—
193#386—22no—
194#386—24no—
195#386—24no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—24no—
200#384—21no—
201#384—21no—
202#384C2312yes—
203#382—21no—
204#382—21no—
205#382—22no—
206#382C2312yes—
207#382C2312yes—
208#382C2312yes—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—24no—
212C4×C4⋊C4—24no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2314yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—24no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4D414yes—
222#376C2312yes—
223C2×#95—22no—
224C43C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236#336C2312yes—
237C25—21no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41C2×Q814yes—
244C22×Q8—22no—
245C22×Q8C2411yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—24no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4C2×Q814yes—
255C22×D4C2412yes—
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C4⋊Q8—24no—
283C4⋊Q8—24no—
284#127—24no—
285#127C2×Q818yes—
286#125—24no—
287#125—24no—
288#124—24no—
289#124C2×Q818yes—
290C22⋊Q8—216no—
291#121—216no—
292#120—216no—
293C4×Q8—216no—
294C4×Q8—42no—
295C4×Q8—42no—
296C4×D4—42no—
297C4×D4—42no—
298C4×D4—42no—
299C4×D4—42no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4—48no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—88no—
306#117—22no—
307#117—22no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—44no—
313#117C2×Q814yes—
314#117#4114yes—
315#117#4114yes—
316#117#4114yes—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—48no—
356C2×C22⋊C4—84no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42C2×D412yes—
368C2×C42C2×D414yes—
369C2×C42C2×D414yes—
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2412yes—
374#107—44no—
375#107—42no—
376#107—42no—
377#106—44no—
378#106—42no—
379#106—42no—
380C4⋊C8—28no—
381C4⋊C8—28no—
382#103—24no—
383#103—24no—
384#103—42no—
385#103—42no—
386#102—24no—
387#102—24no—
388#102—28no—
389#102—28no—
390#102—44no—
391#102—44no—
392#102—42no—
393#102—42no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95C2×D414yes—
404#95C2×D414yes—
405C24—22no—
406C24—21no—
407C24—24no—
408C24—42no—
409C24—42no—
410C24—82no—
411C24—82no—
412C24C22×D411yescharacteristic
413#41—28no—
414#41—28no—
415#41—44no—
416#41—44no—
417C2×Q8—28no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—82no—
424C2×Q8—82no—
425C2×Q8C22×Q812yes—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—216no—
429C2×D4—28no—
430C2×D4—28no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—84no—
448C2×D4—84no—
449C2×D4C22×Q814yes—
450C2×D4C22×Q812yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D414yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×Q812yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4C2511yescharacteristic
490M16—44no—
491M16—44no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—44no—
505C2×C8—44no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4C22⋊Q818yes—
531C4⋊C4C2×#4114yes—
532C4⋊C4C2×#4114yes—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—216no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—48no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—82no—
544C22⋊C4—84no—
545C22⋊C4C22⋊Q818yes—
546C42—24no—
547C42—24no—
548C42—24no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—22no—
554C42—22no—
555C42—44no—
556C42—44no—
557C42—44no—
558C42—42no—
559C42—84no—
560C42—84no—
561C42C22×Q814yes—
562C42C2×#4114yes—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—22no—
567C23—42no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—41no—
572C23—42no—
573C23—41no—
574C23—42no—
575C23—48no—
576C23—88no—
577C23—162no—
578C23—162no—
579C23—164no—
580C23C2×C22⋊Q812yes—
581C23#52512yes—
582C23#53512yes—
583C23C23×D411yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—41no—
587Q8—41no—
588Q8—88no—
589D4—24no—
590D4—24no—
591D4—22no—
592D4—22no—
593D4—48no—
594D4—48no—
595D4—42no—
596D4—42no—
597D4—41no—
598D4—41no—
599D4—816no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—84no—
633C2×C4—164no—
634C2×C4—164no—
635C2×C4—164no—
636C2×C4—162no—
637C2×C4—162no—
638C2×C4—162no—
639C2×C4C2×C22⋊Q818yes—
640C2×C4C2×C22⋊Q812yes—
641C2×C4#52512yes—
642C2×C4#53512yes—
643C2×C4#55414yes—
644C2×C4#55414yes—
645C2×C4#55914yes—
646C2×C4#55914yes—
647C2×C4C23×Q811yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C8—42no—
654C8—42no—
655C8—42no—
656C8—42no—
657C22—24no—
658C22—22no—
659C22—22no—
660C22—24no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22—48no—
665C22—82no—
666C22—81no—
667C22—81no—
668C22—82no—
669C22—82no—
670C22—84no—
671C22—168no—
672C22#258014yes—
673C22C22×C22⋊Q811yescharacteristic
674C22C2×#52511yescharacteristic
675C22C2×#53511yescharacteristic
676C22#358412yes—
677C22#358512yes—
678C4—28no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—44no—
686C4—82no—
687C4—81no—
688C4—82no—
689C4—82no—
690C4—81no—
691C4#258014yes—
692C4C2×#55411yescharacteristic
693C4C2×#55411yescharacteristic
694C4C2×#55911yescharacteristic
695C4C2×#55911yescharacteristic
696C2—24no—
697C2—82no—
698C2—84no—
699C2C2×#258011yescharacteristic
700C2#6049612yes—
701C2#6058712yes—
702C2#6081911yescharacteristic
703C2#6082711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #34215C2semidirect
3. #29267C2semidirect
4. #29263C2semidirect
5. #29261C2semidirect
6. #29256C2semidirect
7. #23733C2semidirect
8. C2×#2092C2semidirect
9. #23454C2semidirect
10. #23451C2semidirect
13. #21174C2semidirect
14. #21173C2semidirect
15. #20882C2semidirect
16. #20881C2semidirect
17. #20876C2semidirect
18. #20876C2semidirect
19. #20754C2semidirect
44. #2331C22semidirect
51. C4×#125C22semidirect
59. #2097C22semidirect
60. #2097C22semidirect
63. #2092C22semidirect
64. #2092C22semidirect
65. #2026C22semidirect
67. #2025C22semidirect
68. #2024C22semidirect
70. #2023C22semidirect
75. #1949C22semidirect
76. #1949C22semidirect
77. #1948C22semidirect
78. #1947C22semidirect
79. #1865C22semidirect
82. #1864C22semidirect
83. #1863C22semidirect
86. #1862C22semidirect
87. #1832C22semidirect
88. #1831C22semidirect
104. #535Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#342752
#342152
#292672
#292632
#292612
#292562
#237332
C2×#20922
#234542
#234512
#212632
#211742
#211732
#208822
#208812
#208762
#207542